3.6. If F is a ﬁeld, prove that its only ideals are (0) and F itself.

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Expert's answer

2022-06-19T17:51:56-0400

Let "I" be an ideal of a ﬁeld "F". Assume that "(0)\\ne I" and let "0\\ne a\\in I". Then as "F" is a field, there exists "a^{-1}\\in F" such that "a^{-1}a=1". Hence "1\\in I" and "I=F". So every non-zero ideal if "F" is equal to "F".

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